Application of Bayesian Method

Machine Company manufactures pressure-relief valves and they have determined that 0.1% of valves they produce are defective due to critical flaws which affect the material strength.

Due to the criticality of these safety values, non-destructive evaluation (NDE) via radiographic testing is performed on each valve produced by the company to determine whether it is defective. Extensive testing was conducted with a variety of known flaw shapes and sizes to determine the probability of detection (POD) of critical flaws by Machine NDE inspectors. Based on this testing, it was determined that the false positive indication rate was 1%.

If a critical flaw in a pressure-relief valve is detected by one of the NDE inspectors, they will enter a report in the company's problem reporting and tracking system and the valve will be removed from the production line.

Given the data above, find the probability that a valve is defective (actually has flaws), given that an NDE inspector submits a problem report that flaws were detected in the valve.

Assume i)the inspection is 100% correct(ii)the inspection is 95% correct